Homology is a fundamental tool in algebra and topology, used to study mathematical objects through algebraic invariants. These invariants are often difficult to compute directly. A common strategy for computing the homology of objects that arise naturally in families is to analyze their behavior in a stable range.
One speaks of homological stability when, in a family indexed by a parameter n, the homology groups in degree d become independent of n once n is sufficiently large relative to d. The part of the homology that stabilizes is called stable homology. This stable part is often more accessible than the unstable one. By combining computations of stable homology with homological stability results, one can obtain explicit descriptions, of homology for large objects in the family.
Such phenomena occur in many natural settings.
In this talk, I will discuss the homology of groups. Classical examples of families exhibiting homological stability include the symmetric groups and the braid groups, which illustrate the strength and ubiquity of this phenomenon. I will first introduce the general notions of homological stability and stable homology, and then turn to the case of twisted coefficients, where stable homology is closely related to functor homology. I will then focus on the family of groups Aut(Fn) and present recent computations of functor homology obtained in joint work with Minkyu Kim.
